We will mostly concern ourselves with the technique used by Belishev in his 2003 paper [1]} called the Boundary Control (BC) method. This technique utilizes the classical Gelfand representation for commutative Banach algebras. That is, for a surface \(S\) , the spectrum (maximal ideal space) of the commutative Banach algebra of holomorphic functions is homeomorphic to \(S\) . Alongside this result, Belishev then used the complex structure of the algebra to determine the metric up to conformal equivalence. Furthermore, it is the Gelfand transform that allows for all of this information to be obtained from the boundary. This led to a solution to the Calderón problem in dimension 2. For more on the boundary control method, see [2]}.
